Simplify the following expression and state the condition under which the simplification is valid: $p = \dfrac{q^2 - 7q}{q^2 - 16q + 63}$
Answer: First factor the expressions in the numerator and denominator. $ \dfrac{q^2 - 7q}{q^2 - 16q + 63} = \dfrac{(q)(q - 7)}{(q - 9)(q - 7)} $ Notice that the term $(q - 7)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(q - 7)$ gives: $p = \dfrac{q}{q - 9}$ Since we divided by $(q - 7)$, $q \neq 7$. $p = \dfrac{q}{q - 9}; \space q \neq 7$